cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A246437 Expansion of (1/2)*(1/(x+1)+1/(sqrt(-3*x^2-2*x+1))).

Original entry on oeis.org

1, 0, 2, 3, 10, 25, 71, 196, 554, 1569, 4477, 12826, 36895, 106470, 308114, 893803, 2598314, 7567465, 22076405, 64498426, 188689685, 552675364, 1620567764, 4756614061, 13974168191, 41088418150, 120906613076, 356035078101, 1049120176954
Offset: 0

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Author

Vladimir Kruchinin, Nov 14 2014

Keywords

Comments

a(2101) has 1001 decimal digits. - Michael De Vlieger, Apr 25 2016
This is the analog for Coxeter type B of Motzkin sums (A005043) for Coxeter type A, see the article by Athanasiadis and Savvidou. - F. Chapoton, Jul 20 2017
Number of compositions of n into exactly n nonnegative parts avoiding part 1. a(4) = 10: 0004, 0022, 0040, 0202, 0220, 0400, 2002, 2020, 2200, 4000. - Alois P. Heinz, Aug 19 2018
From Peter Bala, Jan 07 2022: (Start)
The binary transform is A088218. The inverse binomial transform is a signed version of A027306 and the second inverse binomial transform is a signed version of A027914.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

Links

Crossrefs

Cf. A002426, A005043, A027306, A027914, A088218, A104507, A370616, A386548.

Programs

  • Mathematica
    CoefficientList[Series[(1/2) (1 / (x + 1) + 1 / (Sqrt[-3 x^2 - 2 x + 1])), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
Table[(-1)^n (Hypergeometric2F1[1/2, -n, 1, 4] + 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)
Table[Sum[Binomial[n, k] Binomial[n - k - 1, n - 2 k], {k, 0, n/2}], {n, 0, 28}] (* Michael De Vlieger, Apr 25 2016 *)
  • Maxima
    a(n):=sum(binomial(n,k)*binomial(n-k-1,n-2*k),k,0,n/2);
  • Sage
    def a(n):
    if n < 3: return [1,0,2][n]
    return n*hypergeometric([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4)
[simplify(a(n)) for n in (0..28)] # Peter Luschny, Nov 14 2014

Formula

a(n) = Sum_{k = 0..n/2} binomial(n,k)*binomial(n-k-1,n-2*k).
A(x) = 1 + x*B'(x)/B(x), where B(x) = (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)) is the o.g.f. of A005043.
a(n) = n*hypergeom([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4) for n>=3. - Peter Luschny, Nov 14 2014
a(n) ~ 3^(n+1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 14 2014
a(n) = (-1)^n*(hypergeom([1/2, -n], [1], 4) + 1)/2. - Vladimir Reshetnikov, Apr 25 2016
D-finite with recurrence: n*(a(n) - a(n-1)) = (5*n-6)*a(n-2) + 3*(n-2)*a(n-3). - Vladimir Reshetnikov, Oct 13 2016
a(n) = [x^n]( (1 - x + x^2)/(1 - x) )^n. - Peter Bala, Jan 07 2022

A104507 Row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.

Original entry on oeis.org

1, 0, -2, -3, 2, 15, 19, -28, -134, -129, 353, 1254, 791, -4238, -11818, -3123, 49162, 110007, -17783, -554458, -996323, 690932, 6096792, 8624747, -12287153, -65419110, -69285296, 178655307, 684550946, 483569751, -2354830741, -6970706252, -2324044054, 29195280375, 68793790705
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2005

Keywords

Crossrefs

Cf. A104505, A1086223, A246437, A370616, A386548.

Programs

  • Mathematica
    CoefficientList[Series[(x/((1 - x)) + 1/((-Sqrt[5 x^2 - 2 x + 1] + x + 1)) x (1 - (5 x - 1)/(Sqrt[5 x^2 - 2 x + 1]))), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 05 2015 *)
  • Maxima
    a(n):=sum((-1)^j*binomial(n,j)*binomial(n-j-1,n-2*j),j,0,n/2); /* Vladimir Kruchinin, Oct 04 2015 */
  • PARI
    a(n)=sum(k=0,n,polcoeff((1+x-x^2)^n,n+k))
  • PARI
    a(n) = sum(k=0, n/2, (-1)^k*binomial(n,k)*binomial(n-k-1,n-2*k));
vector(40, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Formula

G.f.: (x/((1-x))+1/((-sqrt(5*x^2-2*x+1)+x+1))*x*(1-(5*x-1)/(sqrt(5*x^2-2*x+1)))). - Vladimir Kruchinin, Oct 04 2015
a(n) = Sum_{j=0..n/2}((-1)^j*binomial(n,j)*binomial(n-j-1,n-2*j)). - Vladimir Kruchinin, Oct 04 2015
From Peter Bala, Jul 24 2025: (Start)
a(n) = [x^n] (1 - x^2/(1 - x))^n. Cf. A246437.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all primes p and all positive integers n and k.
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 - x^2 - x^3 + x^4 + 4*x^5 + ... is the g.f. of A108623.(End)

A370616 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^n.

Original entry on oeis.org

1, 0, 2, 3, 14, 35, 125, 371, 1238, 3909, 12847, 41580, 136577, 447187, 1473341, 4855703, 16053830, 53138243, 176233967, 585202261, 1945964079, 6478043120, 21588979876, 72016891508, 240452892569, 803489258285, 2686964354375, 8991840800136, 30110638705889
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2024

Keywords

Crossrefs

Cf. A046736, A213684, A104507, A246437, A386548.

Programs

  • Mathematica
    Table[Sum[Binomial[-1 - k + n, -2*k + n] Binomial[-1 + k + n, k], {k, 0, n/2}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 30 2025 *)
  • PARI
    a(n, s=2, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(n-k-1,n-2*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x-x^2) / (1-x) ).
From Peter Bala, 26 Jul 2025: (Start)
a(n) = n * hypergeom([1 + n, 1 - n/2, 3/2 - n/2], [2, 2 - n], -4) for n >= 3.
P-recursive: 5*n*(74*n^3-493*n^2+1075*n-766)*(n-1)*a(n) = 2*(n-1)*(296*n^4-2120*n^3+5393*n^2-5716*n+2100)*a(n-1) + 2*(1184*n^5-10256*n^4+34088*n^3-53995*n^2+40397*n-11250)*a(n-2) - 2*(n-3)*(2*n-5)*(74*n^3-271*n^2+311*n-110)*a(n-3) with a(0) = 1, a(1) = 0 and a(2) = 2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. (End)
a(n) ~ sqrt(1/12 + sqrt(10/37)*(sin(arcsin((13*sqrt(37/10))/40)/3)/3)) * (8*((1 + sqrt(34)*cos(arccos(2461/(1088*sqrt(34)))/3))/15))^n / sqrt(Pi*n). - Vaclav Kotesovec, Jul 30 2025
Showing 1-3 of 3 results.

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